Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions

Abstract

In this paper, we focus on strongly local regular Dirichlet forms, especially those satisfying Morrey-type inequalities. We prove the equivalence between resistance estimates and heat kernel estimates in this case. Self-similar forms on fractals serve as a major application, where we construct a spatially inhomogeneous scaling function and characterize all the doubling self-similar measures. Further, on some special examples, the resistance conditions are reduced to some geometric conditions, on which a complete theory on self-similar Dirichlet spaces is established therein. In particular, we construct a concrete example on rotated triangle fractals, where the optimal heat kernel estimate is not related at all to the lower scaling exponent.

Figures (9)

• Figure 1: The ball $B(x,C_1\bar{R}r)$ intersects at most $L_1$ cells from the partition $\Lambda_{\mathbf{{ \if@compatibility \mathchar"0112 {} \mathchar"0112 }}}(r)$.
• Figure 2: Two points $x$ and $y$ are connected by a sequence $\{K_{w^{(i)}}\}_{i=1}^{L_{2}}$ with $x\in K_{w^{(1)}}, y\in K_{w^{(L_2)}}$ and $K_{w^{(i-1)}}\cap K_{w^{(i)}}\neq\emptyset$, where $w^{(i)}\in\Lambda_{\mathbf{{ \if@compatibility \mathchar"0112 {} \mathchar"0112 }}}(r)$.
• Figure 3: An illustration of $\Gamma_x^{(i)}$.
• Figure 4: An illustration of $K_x$.
• Figure 5: ${ \if@compatibility \mathchar"0115 {} \mathchar"0115 }=1/4$.

Theorems & Definitions (99)

• Definition 1: Tail estimate of heat semigroup
• Definition 2: Morrey-type inequality
• Proposition 2.1
• proof
• Definition 3: On-diagonal upper estimate
• Definition 4: Upper estimate with exponential tail
• Definition 5: Near-diagonal lower estimate
• Definition 6: Lower estimate with exponential tail
• Proposition 2.2
• proof : Sketch of the proof